1. ## Sets and counting

If A union B = an empty set which of the following are true? (give all correct choices)
a. A is a proper set of B
b. B is a proper set of A
c. A intersection B = an empty set (correct)
d. A = an empty set (Correct)
e. B = an empty set (Correct)

Plese let me know if I have picked all the correct choices and if not explain why, because my textbook does not explain this very well.

2. Originally Posted by d.darbyshire
If A union B = an empty set which of the following are true? (give all correct choices)
a. A is a proper set of B
b. B is a proper set of A
c. A intersection B = an empty set (correct)
d. A = an empty set (Correct)
e. B = an empty set (Correct)

Plese let me know if I have picked all the correct choices and if not explain why, because my textbook does not explain this very well.
Looks OK to me the condition implies A and B are both the empty set, so
cases c thru e are true and a and b are false.

RonL

3. Originally Posted by d.darbyshire
If A union B = an empty set which of the following are true? (give all correct choices)
a. A is a proper set of B
b. B is a proper set of A
c. A intersection B = an empty set (correct)
d. A = an empty set (Correct)
e. B = an empty set (Correct)

Plese let me know if I have picked all the correct choices and if not explain why, because my textbook does not explain this very well.
If $A\cup B=\{\}$ then both $A,B$ must be empty sets. Because if $A,B$ are not then there is an element of $A \cup B$, but then it is also an element of $\{\}$ (by definition of equality). But this is a contradiction because $\{\}$ has no elements. Thus, both $A \mbox{ and } B=\{\}$.

A proper subset of a set is subset of a set which is not equal to that set. In mathematical terms,
$P\subset S\mbox{ iff } P\subseteq S,P\not =S$. Since $A=B$ as demonstrated in the previous paragraph they cannot be proper subsets of eachother. Thus, answer to #1 and #2 is no.

Assume $A\cap B$ is not empty. Then there exists an element which is common to both $A\mbox{ and }B$(by definition of intersection) which is not possible because $A,B$ are both empty. Thus, the answer to #3 is yes.

Questions #4 and #5 were demonstrated in the first paragraph, that they are both empty.

4. A question on notation here. The book I learned set theory from has the statement $A \subset B$ defined as "A is a subset of B" implying that all elements of A are contained in B. This means that $A=B$ is a possibility. However, I've noted on a number of occasions that members of the forum are using $A \subset B$ to mean that A cannot equal B. Such as in the previous post:
$P \subset S\mbox{ iff } P\subseteq S,P\not=S$
Is my book using non-standard notation, or are there different conventions in use?

-Dan

5. Originally Posted by topsquark
A question on notation here. The book I learned set theory from has the statement $A \subset B$ defined as "A is a subset of B" implying that all elements of A are contained in B. This means that $A=B$ is a possibility. However, I've noted on a number of occasions that members of the forum are using $A \subset B$ to mean that A cannot equal B. Such as in the previous post:

Is my book using non-standard notation, or are there different conventions in use?

-Dan
Look it up on Wikipedia!
http://en.wikipedia.org/wiki/Subset
Yes, I believe that your book is wrong (or non-standard) if your memory is correct.

6. Originally Posted by topsquark
A question on notation here. The book I learned set theory from has the statement $A \subset B$ defined as "A is a subset of B" implying that all elements of A are contained in B. This means that $A=B$ is a possibility. However, I've noted on a number of occasions that members of the forum are using $A \subset B$ to mean that A cannot equal B. Such as in the previous post:

Is my book using non-standard notation, or are there different conventions in use?

-Dan
Conventions change with time and from author to author. In general you
have to look at what an author has defined their symbols to mean.

However, having said that since we have both symbols $\subset$ and $\subseteq$ it seems
silly not to take advantage and let them denote proper subset, and subset
which takes advantage of the analogy with $>$ and $\ge$.

RonL

7. Originally Posted by ThePerfectHacker
Look it up on Wikipedia!
http://en.wikipedia.org/wiki/Subset
Yes, I believe that your book is wrong (or non-standard) if your memory is correct.
I don't mean to say anything bad about Wikipedia, but as it is written by users (aka not professionally written) I don't take anything I see there as gospel. I'm not saying that anything there has been deliberately written incorrectly, and it definately IS useful, but I don't know what kind of checks they use to make sure mistakes don't get in. I HAVE heard that errors are in there. So I like to check other places for verification.

-Dan

8. Originally Posted by topsquark
I don't mean to say anything bad about Wikipedia, but as it is written by users (aka not professionally written) I don't take anything I see there as gospel. I'm not saying that anything there has been deliberately written incorrectly, and it definately IS useful, but I don't know what kind of checks they use to make sure mistakes don't get in. I HAVE heard that errors are in there. So I like to check other places for verification.

-Dan
Just because a person does not have a Ph.D. in math does not mean it is professional, some amatuers know much more than professors.

9. Originally Posted by ThePerfectHacker
Just because a person does not have a Ph.D. in math does not mean it is professional, some amatuers know much more than professors.
When I used the phrase "not professionally written" I was referring to the fact that my 10 year old niece can post on Wikipedia. That's nice, but is someone going to go through her post and correct any errors? I don't know who is running the site and what kind of editing they do. Until I find out I will regard any information from the site as suspect.

I was most certainly not taking a poke at those without PhDs, as I don't have one either.

-Dan